A Ham Sandwich Analogue for Quaternionic Measures and Finite Subgroups of S^3

Abstract

A "ham sandwich" theorem is established for n quaternionic Borel measures on quaternionic space H^n. For each finite subgroup G of S^3, it is shown that there is a quaternionic hyperplane H and a corresponding tiling of H^n into |G| fundamental regions which are rotationally symmetric about H with respect to G, and satisfy the condition that for each of the n measures, the "G average" of the measures of these regions is zero. If each quaternionic measure is a 4-tuple of finite Borel measures on R^{4n}, the original ham sandwich theorem on R^{4n} is recovered when G = Z_2. The theorem applies to [n/4] finite Borel measures on R^n, and when G is the quaternion group Q_8 this gives a decomposition of R^n into 2 rings of 4 cubical "wedges" each, such that the measure any two opposite wedges is equal for each finite measure.